Solve for x and simplify the answer fully. \[ \frac{x+3}{x-1}=\frac{5}{9} \]

To solve for \( x \) in the equation \[ \frac{x+3}{x-1}=\frac{5}{9}, \] we can start by cross-multiplying: \[ 9(x + 3) = 5(x - 1). \] Now, distribute both sides: \[ 9x + 27 = 5x - 5. \] Next, we can isolate \( x \) by moving the terms involving \( x \) to one side and the constant terms to the other side: \[ 9x - 5x = -5 - 27, \] which simplifies to: \[ 4x = -32. \] Now, divide both sides by 4: \[ x = -8. \] Therefore, the solution is \[ \boxed{-8}. \]